publications

2026

1. diffSPH: Differentiable smoothed particle hydrodynamics for hybrid machine learning solutions in fluid mechanics

   Rene Winchenbach, and Nils Thuerey

   Journal of Computational Physics, 2026

   Abs DOI arXiv Bib Code

   We present diffSPH, a novel open-source differentiable Smoothed Particle Hydrodynamics (SPH) framework developed entirely in PyTorch with GPU acceleration. diffSPH is designed centrally around differentiation to facilitate optimization and machine learning (ML) applications in Computational Fluid Dynamics (CFD), including training neural networks and the development of hybrid models. Its differentiable SPH core, and schemes for compressible (with shock capturing and multi-phase flows), weakly compressible (with boundary handling and free- surface flows), and incompressible physics, enable a broad range of application areas. We demonstrate the framework’s unique capabilities through several applications, including addressing particle shifting via a novel, target- oriented approach by minimizing physical and regularization loss terms, a task often intractable in traditional solvers. Further examples include optimizing initial conditions and physical parameters to match target trajectories, shape optimization, implementing a solver-in-the-loop setup to emulate higher-order integration, and demonstrating gradient propagation through hundreds of full simulation steps. Prioritizing readability, usability, and extensibility, this work offers a foundational platform for the CFD community to develop and deploy novel neural networks and adjoint optimization applications.

   @article{winchenbach2026diffsph,
     author = {Winchenbach, Rene and Thuerey, Nils},
     title = {diffSPH: Differentiable smoothed particle hydrodynamics for hybrid
                       machine learning solutions in fluid mechanics},
     journal = {Journal of Computational Physics},
     volume = {555},
     pages = {114769},
     year = {2026},
     doi = {10.1016/j.jcp.2026.114769},
     url = {https://doi.org/10.1016/j.jcp.2026.114769},
   }

2. Solving boundary handling analytically in two dimensions for smoothed particle hydrodynamics

   Rene Winchenbach, and Andreas Kolb

   Journal of Computational Physics, 2026

   Abs DOI arXiv Bib

   We present a fully analytic approach for evaluating boundary integrals in two dimensions for Smoothed Particle Hydrodynamics (SPH). Conventional methods often rely on boundary particles or wall re-normalization approaches derived from applying the divergence theorem, whereas our method directly evaluates the area integrals for SPH kernels and gradients over triangular boundaries. This direct integration strategy inherently accommodates higher-order boundary conditions, such as piecewise cubic fields defined via Finite Element stencils, enabling analytic and flexible coupling with mesh-based solvers. At the core of our approach is a general solution for compact polynomials of arbitrary degree over triangles by decomposing the boundary elements into elementary integrals that can be solved with closed-form solutions. We provide a complete, closed-form solution for these generalized integrals, derived by relating the angular components to Chebyshev polynomials and solving the resulting radial integral via a numerically stable evaluation of the Gaussian hypergeometric function _2F_1. Our solution is robust and adaptable and works regardless of triangle geometries and kernel functions. We validate the accuracy against high- precision numerical quadrature rules, as well as in problems with known exact solutions. We provide an open-source implementation of our general solution using differentiable programming to facilitate the adoption of our approach to SPH and other contexts that require analytic integration over polygonal domains. Our analytic solution outperforms existing numerical quadrature rules for this problem by up to five orders of magnitude, for integrals and their gradients, while providing a flexible framework to couple arbitrary triangular meshes analytically to Lagrangian schemes, building a strong foundation for addressing several grand challenges in SPH and beyond.

   @article{winchenbach2026boundary,
     author = {Winchenbach, Rene and Kolb, Andreas},
     title = {Solving boundary handling analytically in two dimensions for smoothed
                       particle hydrodynamics},
     journal = {Journal of Computational Physics},
     volume = {555},
     pages = {114788},
     year = {2026},
     doi = {10.1016/j.jcp.2026.114788},
     url = {https://doi.org/10.1016/j.jcp.2026.114788},
   }

2025

1. MoriNet - Machine Learning from a Mori-Zwanzig Perspective

   Rene Winchenbach, and Nils Thuerey

   In 19th International SPHERIC Workshop, Barcelona, Spain, 2025

2024

1. 

   Lipschitz-agnostic, efficient and accurate rendering of implicit surfaces

   Rene Winchenbach, Michael Möller, and Andreas Kolb

   The Visual Computer, 2024

   Abs DOI Bib

   In this paper, we propose an accurate and controllable rendering process for implicit surfaces with no or unknown analytic Lipschitz constants. Our process is built upon a ray-casting approach where we construct an adaptive Chebyshev proxy along each ray to perform an accurate intersection test via a robust and multi-stage searching method. By taking into account approximation errors and numerical conditions, our methods comprise several pre-conditioning and post-processing stages to improve the numerical accuracy, which potentially applied recursively. The intersection search is performed by evaluating a QR decomposition on the Chebyshev proxy function, which can be done in a numerically accurate way. Our process achieves comparable accuracy to other techniques that impose more constraints on the surface, e.g., knowledge of Lipschitz constants, and higher accuracy compared to approaches that impose similar constraints as our approach.

   @article{winchenbach2024lipschitz,
     author = {Winchenbach, Rene and M{\"o}ller, Michael and Kolb, Andreas},
     title = {Lipschitz-agnostic, efficient and accurate rendering of implicit surfaces},
     journal = {The Visual Computer},
     volume = {40},
     number = {11},
     pages = {7925--7944},
     year = {2024},
     publisher = {Springer},
     doi = {10.1007/s00371-023-03216-y},
     url = {https://doi.org/10.1007/s00371-023-03216-y},
   }

2. 

   Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics

   Rene Winchenbach, and Nils Thuerey

   In 12th International Conference on Learning Representations, ICLR 2024, Vienna, Austria, 2024

   Abs arXiv Bib Code Website

   Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

   @inproceedings{Winchenbach2024SFBC,
     author = {Winchenbach, Rene and Thuerey, Nils},
     title = {Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics},
     booktitle = {12th International Conference on Learning Representations, {ICLR} 2024,
                       Vienna, Austria},
     year = {2024},
     url = {https://openreview.net/forum?id=HKgRwNhI9R},
   }

3. Physically-Motivated Machine Learning Models for Lagrangian Fluid Mechanics

   Rene Winchenbach, and Nils Thuerey

   In 18th International SPHERIC Workshop, Berlin, Germany, 2024

   Code
4. Cross-Validation of SPH-based Machine Learning Models using the Taylor-Green Vortex Case

   Rene Winchenbach, and Nils Thuerey

   In Particle Methods and Applications Conference, Santa Fe, USA, 2024

2023

1. Evaluation of particle-based smoothed particle hydrodynamics boundary handling approaches in computer animation

   Rustam Akhunov, Rene Winchenbach, and Andreas Kolb

   Comput. Animat. Virtual Worlds, 2023

   Abs DOI Bib

   Boundary handling is an important aspect of fluid simulation, and several boundary handling approaches exist in smoothed particle hydrodynamics (SPH), which have individual strengths and weaknesses. However, comparing different boundary handling approaches is challenging as there is no common basis for evaluations, that is, no universal set of experiments with quantitative evaluation across different methods, especially within computer animation where many evaluations rely mainly on visual perception. This article proposes a set of experiments to aid the evaluation of the main categories of fluid‐boundary interactions that are important in computer animation, that is, no motion (resting) fluid, tangential and normal motion of a fluid with respect to the boundary, and a fluid impacting a corner. We propose ten experiments, comprising experimental setup and quantitative evaluation with optional visual inspections, that are arranged in four groups which focus on one of the main category of fluid‐boundary interactions. We use these experiments to evaluate three particle‐based boundary handling methods, that is, pressure mirroring, pressure boundaries, and moving least squares pressure extrapolation, in combination with two incompressible SPH fluid simulation methods, namely IISPH and DFSPH, to establish a quantifiable relation between different combinations of boundary handling with simulation approaches and the main categories of fluid‐boundary interactions. Finally, we summarize all results in a rating table and show how our experiments can be used to determine the promising method for specific requirements regarding a given constellation of fluid‐boundary interaction.

   @article{DBLP:journals/jvca/AkhunovWK23,
     author = {Akhunov, Rustam and Winchenbach, Rene and Kolb, Andreas},
     title = {Evaluation of particle-based smoothed particle hydrodynamics boundary
                       handling approaches in computer animation},
     journal = {Comput. Animat. Virtual Worlds},
     volume = {34},
     number = {6},
     year = {2023},
     doi = {10.1002/cav.2138},
     url = {https://doi.org/10.1002/cav.2138},
   }

2. 

   A Hybrid Framework for Fluid Flow Simulations: Combining SPH with Machine Learning

   Rene Winchenbach, and Nils Thuerey

   In 17th International SPHERIC Workshop, Rhodes, Greece, 2023

2022

1. 

   Visualizing Optimizers using Chebyshev Proxies and Fatou Sets

   Rene Winchenbach, and Nils Thuerey

   In VMV 2022, 27th International Symposium on Vision, Modeling, and Visualization, Konstanz, Germany, September 27-30, 2022, 2022

   Abs DOI Bib

   With recent advances in optimization many different optimization approaches have been proposed, especially regarding the optimization of weights for neural networks. However, comparing these approaches in a visually succinct and intuitive manner is difficult to do, especially without relying on simplified toy examples that may not be representative. In this paper, we present a visualization toolkit using a modified variant of Fatou sets of functions in the complex domain to directly visualize the convergence behavior of an optimizer across a large range of input values. Furthermore, we propose an approach of generating test functions based on polynomial Chebyshev proxies, with polynomial degrees up to 11217, and a modification of these proxies to yield functions that are strictly positive with known global minima, i.e., roots. Our proposed toolkit is provided as a cross platform open source framework in C++ using OpenMP for parallelization. Finally, for menomorphic functions the process generates visually interesting fractals, which might also be interesting from an artistic standpoint.

   @inproceedings{DBLP:conf/vmv/WinchenbachT22,
     author = {Winchenbach, Rene and Thuerey, Nils},
     title = {Visualizing Optimizers using Chebyshev Proxies and Fatou Sets},
     booktitle = {{VMV} 2022, 27th International Symposium on Vision, Modeling, and
                       Visualization, Konstanz, Germany, September 27-30, 2022},
     pages = {75--82},
     publisher = {Eurographics Association},
     year = {2022},
     doi = {10.2312/vmv.20221206},
     url = {https://doi.org/10.2312/vmv.20221206},
   }

2021

1. Spatially adaptive Smoothed Particle Hydrodynamics

   Rene Winchenbach

   Siegen University, Germany, 2021

   DOI Bib

   @phdthesis{DBLP:phd/basesearch/Winchenbach21,
     author = {Winchenbach, Rene},
     title = {Spatially adaptive Smoothed Particle Hydrodynamics},
     school = {Siegen University, Germany},
     year = {2021},
     doi = {10.25819/UBSI/10084},
     url = {https://dspace.ub.uni-siegen.de/handle/ubsi/2172},
   }

2020

1. 

   Multi-Level Memory Structures for Simulating and Rendering Smoothed Particle Hydrodynamics

   Rene Winchenbach, and Andreas Kolb

   Comput. Graph. Forum, 2020

   Abs DOI Bib

   In this paper, we present a novel hash map‐based sparse data structure for Smoothed Particle Hydrodynamics, which allows for efficient neighbourhood queries in spatially adaptive simulations as well as direct ray tracing of fluid surfaces. Neighbourhood queries for adaptive simulations are improved by using multiple independent data structures utilizing the same underlying self‐similar particle ordering, to significantly reduce non‐neighbourhood particle accesses. Direct ray tracing is performed using an auxiliary data structure, with constant memory consumption, which allows for efficient traversal of the hash map‐based data structure as well as efficient intersection tests. Overall, our proposed method significantly improves the performance of spatially adaptive fluid simulations and allows for direct ray tracing of the fluid surface with little memory overhead.

   @article{DBLP:journals/cgf/WinchenbachK20,
     author = {Winchenbach, Rene and Kolb, Andreas},
     title = {Multi-Level Memory Structures for Simulating and Rendering Smoothed
                       Particle Hydrodynamics},
     journal = {Comput. Graph. Forum},
     volume = {39},
     number = {6},
     pages = {527--541},
     year = {2020},
     doi = {10.1111/cgf.14090},
     url = {https://doi.org/10.1111/cgf.14090},
   }

2. 

   Semi-analytic boundary handling below particle resolution for smoothed particle hydrodynamics

   Rene Winchenbach, Rustam Akhunov, and Andreas Kolb

   ACM Trans. Graph., 2020

   Abs DOI Bib

   In this paper, we present a novel semi-analytical boundary handling method for spatially adaptive and divergence-free smoothed particle hydrodynamics (SPH) simulations, including two-way coupling. Our method is consistent under varying particle resolutions and allows for the treatment of boundary features below the particle resolution. We achieve this by first introducing an analytic solution to the interaction of SPH particles with planar boundaries, in 2D and 3D, which we extend to arbitrary boundary geometries using signed distance fields (SDF) to construct locally planar boundaries. Using this boundary-integral-based approach, we can directly evaluate boundary contributions, for any quantity, allowing an easy integration into state of the art simulation methods. Overall, our method improves interactions with small boundary features, readily handles spatially adaptive fluids, preserves particle-boundary interactions across varying resolutions, can directly be implemented in existing SPH methods, and, for non-adaptive simulations, provides a reduction in memory consumption as well as an up to 2× speedup relative to current particle-based boundary handling approaches.

   @article{DBLP:journals/tog/WinchenbachA020,
     author = {Winchenbach, Rene and Akhunov, Rustam and Kolb, Andreas},
     title = {Semi-analytic boundary handling below particle resolution for smoothed
                       particle hydrodynamics},
     journal = {{ACM} Trans. Graph.},
     volume = {39},
     number = {6},
     pages = {173:1--173:17},
     year = {2020},
     doi = {10.1145/3414685.3417829},
     url = {https://doi.org/10.1145/3414685.3417829},
   }

2019

1. 

   Multi-Level-Memory Structures for Adaptive SPH Simulations

   Rene Winchenbach, and Andreas Kolb

   In 24th International Symposium on Vision, Modeling, and Visualization, VMV 2019, Rostock, Germany, September 30 - October 2, 2019, 2019

   Abs DOI Bib

   In this paper we introduce a novel hash map-based sparse data structure for highly adaptive Smoothed Particle Hydrodynamics (SPH) simulations on GPUs. Our multi-level- memory structure is based on stacking multiple independent data structures, which can be created efficiently from the same particle data by utilizing self- similar particle orderings. Furthermore, we propose three neighbor list algorithms that improve performance, or significantly reduce memory requirements, when compared to Verlet-lists for the overall simulation. Overall, our proposed method significantly improves the performance of spatially adaptive methods, allows for the simulation of unbounded domains and reduces memory requirements without interfering with the simulation.

   @inproceedings{DBLP:conf/vmv/Winchenbach019,
     author = {Winchenbach, Rene and Kolb, Andreas},
     title = {Multi-Level-Memory Structures for Adaptive {SPH} Simulations},
     booktitle = {24th International Symposium on Vision, Modeling, and Visualization,
                       {VMV} 2019, Rostock, Germany, September 30 - October 2, 2019},
     pages = {99--107},
     publisher = {Eurographics Association},
     year = {2019},
     doi = {10.2312/vmv.20191323},
     url = {https://doi.org/10.2312/vmv.20191323},
   }

2. Optimized Refinement for Spatially Adaptive SPH

   Rene Winchenbach, and Andreas Kolb

   ACM Trans. Graph., 2019

   Abs DOI Bib

   In this article, we propose an improved refinement process for the simulation of incompressible low-viscosity turbulent flows using Smoothed Particle Hydrodynamics, under adaptive volume ratios of up to 1 : 1, 000, 000. We derive a discretized objective function, which allows us to generate ideal refinement patterns for any kernel function and any number of particles a priori without requiring intuitive initial user-input. We also demonstrate how this objective function can be optimized online to further improve the refinement process during simulations by utilizing a gradient descent and a modified evolutionary optimization. Our investigation reveals an inherent residual refinement error term, which we smooth out using improved and novel methods. Our improved adaptive method is able to simulate adaptive volume ratios of 1 : 1, 000, 000 and higher, even under highly turbulent flows, only being limited by memory consumption. In general, we achieve more than an order of magnitude greater adaptive volume ratios than prior work.

   @article{winchenbach2019optimized,
     author = {Winchenbach, Rene and Kolb, Andreas},
     title = {Optimized Refinement for Spatially Adaptive {SPH}},
     journal = {{ACM} Trans. Graph.},
     volume = {39},
     number = {1},
     pages = {9:1--9:15},
     year = {2019},
     doi = {10.1145/3363555},
     url = {https://doi.org/10.1145/3363555},
   }

2017

1. 

   Infinite continuous adaptivity for incompressible SPH

   Rene Winchenbach, Hendrik Hochstetter, and Andreas Kolb

   ACM Trans. Graph., 2017

   Abs DOI Bib

   In this paper we introduce a novel method to adaptive incompressible SPH simulations. Instead of using a scheme with a number of fixed particle sizes or levels, our approach allows continuous particle sizes. This enables us to define optimal particle masses with respect to, e.g., the distance to the fluid’s surface. A required change in mass due to the dynamics of the fluid is properly and stably handled by our scheme of mass redistribution. This includes temporally smooth changes in particle masses as well as sudden mass variations in regions of high flow dynamics. Our approach guarantees low spatial variations in particle size, which is a core property in order to achieve large adaptivity ratios for incompressible fluid simulations. Conceptually, our approach allows for infinite continuous adaptivity, practically we achieved adaptivity ratios up to 5 orders of magnitude, while still being mass preserving and numerically stable, yielding unprecedented vivid surface detail at comparably low computational cost and moderate particle counts.

   @article{DBLP:journals/tog/WinchenbachHK17,
     author = {Winchenbach, Rene and Hochstetter, Hendrik and Kolb, Andreas},
     title = {Infinite continuous adaptivity for incompressible {SPH}},
     journal = {{ACM} Trans. Graph.},
     volume = {36},
     number = {4},
     pages = {102:1--102:10},
     year = {2017},
     doi = {10.1145/3072959.3073713},
     url = {https://doi.org/10.1145/3072959.3073713},
   }

2016

1. 

   Constrained Neighbor Lists for SPH-based Fluid Simulations

   Rene Winchenbach, Hendrik Hochstetter, and Andreas Kolb

   In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Zurich, Switzerland, July 11-13, 2016, 2016

   Abs DOI Bib

   In this paper we present a new approach to create neighbor lists with strict memory bounds for incompressible Smoothed Particle Hydrodynamics (SPH) simulations. Our proposed approach is based on a novel efficient predictive-corrective algorithm that locally adjusts particle support radii in order to yield neighborhoods of a user-defined maximum size. Due to the improved estimation of the initial support radius, our algorithm is able to efficiently calculate neighborhoods in a single iteration in almost any situation. We compare our neighbor list algorithm to previous approaches and show that our proposed approach can handle larger particle numbers on a single GPU due to its strict guarantees and is able to simulate more particles in real time due to its benefits in regard to performance. Additionally we demonstrate the versatility and stability of our approach in several different scenarios, for example multi-scale simulations and with different kernel functions.

   @inproceedings{DBLP:conf/sca/WinchenbachHK16,
     author = {Winchenbach, Rene and Hochstetter, Hendrik and Kolb, Andreas},
     title = {Constrained Neighbor Lists for {SPH}-based Fluid Simulations},
     booktitle = {Proceedings of the {ACM} SIGGRAPH/Eurographics Symposium on Computer
                       Animation, Zurich, Switzerland, July 11-13, 2016},
     pages = {49--56},
     publisher = {Eurographics Association / {ACM}},
     year = {2016},
     doi = {10.2312/sca.20161222},
     url = {https://doi.org/10.2312/sca.20161222},
   }